Regional Weather and Climate Modeling: Atmospheric Science 820.01 Regional Weather and Climate Modeling: Finite Difference Equations, Vertical Coordinates & Model Grids Keith Hines Byrd Polar Research Center Sources: Wikipedia A.J. Broccoli class notes (http://www.envsci.rutgers.edu/~broccoli/) Peter Lynch class notes (http://mathsci.ucd.ie/met/msc/NWP/MAPH-P313.html) Chris Bretherton class notes (http://www.atmos.washington.edu/2002Q2/547/) Mesoscale Meteorology and Forecasting (Ray) Climate System Modeling (Trenberth) An Introduction to Three-Dimensional Climate Modeling (Washington and Parkinson)

Atmospheric modeling Finite difference Equations Vertical Coordinates Model Grids

Solving the atmospheric equations on an Earth atmosphere grid Numerical Weather Prediction solves for dynamics, kinematics, physics (radiation, latent heat, surface exchange) and hydrology

Finite Differencing Space Time

Use continuous functions rather than grid points? can use multiple sine and cosine waves As we add more waves to the sum more advanced shapes can be represented Finite Element or Galerkin Method

Use continuous functions rather than grid points Cartesian Coordinates: 2D function can be expressed as a sum of as a series of waves (Fourier Series) Differencing with sums of functions is often called “exact” although truncation errors still exist due to the truncated (rather than infinite) series Spherical Coordinates: Use spectral techniques with sums of associated Legendre functions: θ is colatitude, λ is longitude, anm and bnm are normalized harmonic coefficients Rn m and Sn m, are fully normalized spherical harmonics. Pn m (cos θ) is an Associated Legendre Function.

Grid Points – still popular for limited-area models Δ y Represent the atmospheric variables over a rectangular grid. spherical coordinates

How do we solve our differential equations? x0 - 2Δx F(x0) x0 + 2Δx Set up a grid F(x)  F(x0 - Δx) F(x0 + Δx)

Approximate derivative with finite difference

We obtain a centered one What is the error in finite difference? Error is order “h” What if we combine two unbalanced differences? Finite difference approx. from right and left sides We obtain a centered one Centered differences is 2nd order

What if we want a higher order difference scheme? 4th Order Solution: Use more points Above scheme uses 4 adjacent points

Numerical Techniques What do we want in our numerical models? Accuracy (obvious) Balance (not dominated by “noise”) Conservation of key physical properties Numerical Stability (model survives run) Efficiency (obvious)

(early numerical weather prediction) Quasi-Geostrophic Model geostrophic horiz. velocity geostrophic vorticity (curl) Vorticity Equation Thermodynamic Equation Physical Mode + Computational Mode Omega Equation (forces balance between V and T) Or Synoptic Feature + Sound Waves + Gravity Waves How do we achieve balance in a primitive-equation model?

How do we test for computational stability? Can test with simple (linear) advection equation u is a scalar C is advective speed periodic waves sines and cosines Impose simple wave solution Can solve advection equation analytically

F(t)  Could test forward (Euler) time scheme Or test centered (leapfrog) scheme t0 - 2Δt F(t0) t0 + 2Δt F(t)  F(t0 – Δt) F(t0 + Δt)

u = A eiωt Courant Number: Critical for Computational stability Imposed wave solution Courant Number: Critical for Computational stability u = A eiωt μ < 1 for stability (CFL Condition) smaller timestep more stability

Courant Number: Critical for Computational stability Imagine a wave being advected by a speed “c” in a model with grid spacing and time step If a wavelet is advected more than in there can be instability.

Keep your timestep small! u = A eiωt Courant Number: Critical for Computational stability μ < 1 for stability (CFL Condition) smaller timestep more stability Diffusion Equation Stability criteria Keep your timestep small!

Leapfrog scheme Centered space difference (second order) Apply centered differencing in time. Requires 2 previous values to calculate next value. (advection equation)

Numerical solution can cause scalars to improperly become negative False Negatives Red curve “real” solution Blue curve numerical solution Numerical solution can cause scalars to improperly become negative

WRF has an upstream advection scheme, but much more complicated x0 - 2Δx F(x0) x0 + 2Δx U  downstream F(x0 - Δx) F(x0 + Δx) advection - U(x0) [ F(x0) - F(x0 - Δx) ] / Δx for U > 0 - U(x0) [ F(x0 + Δx) - F(x0) ] / Δx for U < 0 WRF has an upstream advection scheme, but much more complicated

Time Integration k1 k4 k2 ,k3 4 Steps to do one step

3rd Order Runge-Kutta time steps used in WRF 3rd order accuracy used with fast acoustic time steps “physics” are outside RK3

Vertical Coordinate Z or P Sigma = P / Ps Intersects topography Terrain following dz/dt, dσ/dt, dp/dt u, v, P, T, q dz/dt, dσ/dt, dp/dt

Modified sigma-coordinate Vertical Velocity Model Levels u, v, T Vertical Velocity terrain-following

Numerical Techniques: Grids

Arakawa Horiz. Staggered Grid Categories Grid A (simplest) Grid B Grid C Grid D Grid E Has advantages for GW dispersion T, P, q (physical scalars) u, v (velocity) Treat gravity waves to simulate desired synoptic waves

Map Projections Mercator Lambert Conformal Polar Stereographic

Map Scale Factor

From WRF Preprocessing System (WPS) Map Parameters &geogrid s_we = 1, e_we = 74, s_sn = 1, e_sn = 61, geog_data_res = '10m','2m', dx = 30000, dy = 30000, map_proj = 'lambert', ref_lat = 34.83 ref_lon = -81.03 truelat1 = 30.0, truelat2 = 60.0, stand_lon = -98.0 geog_data_path = '/data3a/mp/gill/DATA/GEOG' opt_geogrid_tbl_path = 'geogrid/' From WRF Preprocessing System (WPS)

The Hexagon Global Grid? What is next after WRF? The Hexagon Global Grid? http://www.mmm.ucar.edu/wrf/users/workshops/WS2009/presentations/6-06.pdf